TU-1172 Axion dark matter from ﬁrst-order phase transition and very high energy photons from GRB 221009A

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TU-1172

Axion dark matter from ﬁrst-order phase transition,

and very high energy photons from GRB 221009A

Shota Nakagawa,1, ∗Fuminobu Takahashi,1, †Masaki Yamada,1, 2, ‡and Wen Yin1, §

1Department of Physics, Tohoku University, Sendai, Miyagi 980-8578, Japan

2FRIS, Tohoku University, Sendai, Miyagi 980-8578, Japan

(Dated: November 1, 2022)

We study an axion-like particle (ALP) that experiences the ﬁrst-order phase transition with respect to its mass

or potential minimum. This can be realized if the ALP obtains a potential from non-perturbative effects of

SU(N) gauge theory that is conﬁned via the ﬁrst-order phase transition, or if the ALP is trapped in a false vac-

uum at high temperatures until it starts to oscillate about the true minimum. The resulting ALP abundance

is signiﬁcantly enhanced compared to the standard misalignment mechanism, explaining dark matter in a

broader parameter space that is accessible to experiments e.g. IAXO, ALPS-II, and DM-radio. Furthermore,

the viable parameter space includes a region of the mass ma'10−7−10−8eV and the ALP-photon coupling

gaγγ '10−11GeV−1that can explain the recent observation of very high energy photons from GRB221009A

via axion-photon oscillations. The parameter region suggests that the FOPT can generate the gravitational

wave that explains the NANOGrav hint. If the ALP in this region explains dark matter, then the ALP has likely

experienced a ﬁrst-order phase transition.

Introduction. — The Universe has experienced phase

transitions in its thermal history as the temperature de-

creases by many orders of magnitude since the big bang.

The physics of the phase transition can be universally un-

derstood by the behavior of the order parameter. For ex-

ample, the second order phase transition (SOPT) is charac-

terized by a critical exponent which speciﬁes the tempera-

ture dependence of the order parameter. The QCD phase

transition occurring at a temperature around 100MeV is of

this type. The electroweak phase transition proceeds via

the crossover in the Standard Model (SM), where the or-

der parameter changes smoothly. On the contrary, the or-

der parameter changes discontinuously in the ﬁrst-order

phase transition (FOPT), which proceeds via the nucleation

of true-vacuum bubbles. This is realized in many models for

physics beyond the SM. The dynamics of the thermal Uni-

verse drastically changes if one considers a different order

of phase transition. For example, the FOPT results in the

production of gravitational waves (GWs) from the bubble

collisions and the subsequent stochastic dynamics of the

plasma. The production of topological defects, associated

with the spontaneous symmetry breaking, is also modiﬁed

qualitatively.

The QCD axion [1–4] and axion-like particles (ALPs) have

been extensively studied in the literature as candidates for

dark matter (DM) [5–7], and their dynamics is strongly af-

fected by the order of phase transitions. (See for review

[8–14]) The QCD axion acquires a potential from the non-

perturbative effects of QCD. It is temperature-dependent

during the QCD phase transition because the QCD phase

transition is the SOPT. While these axions are expected to

have constant mass from, for example, gravitational instan-

ton effects, they may also acquire a temperature-dependent

effective potential arising from instanton effects of ther-

malized hidden SU(N) gauge sectors. Then, depending on

the order parameter of the conﬁnement phase transition of

SU(N), the ALP may have a temperature-dependent effec-

tive mass during the phase transition. In Ref. [10], the au-

thors considered the case in which the phase transition is

the second order or a crossover, like the QCD. They derived

the upper bound on the ALP abundance, which can be rep-

resented as a lower bound on the ALP decay constant to ex-

plain the DM density.

In this letter, we consider the case in which the ALP ex-

periences the ﬁrst-order phase transition. One of the ex-

amples is the ﬁrst-order phase transition of SU(N) con-

ﬁnement. The phase transition proceeds via the nucle-

ation of true-vacuum bubbles. As the bubble goes through,

the ALP potential suddenly grows within a very short time

scale, and the ALP ﬁeld value does not change much dur-

ing the phase transition. Another example is the so-called

trapped misalignment mechanism [15–17], where the ax-

ion is trapped in a false vacuum at high temperature and

suddenly starts oscillating around the true vacuum when

the potential barrier disappears. In the FOPT case, the re-

sulting ALP abundance is signiﬁcantly enhanced compared

with the SOPT [10] or the standard misalignment mecha-

nism [5–7]. This is because, in the case of FOPT, it is pos-

sible for the ALP to start oscillating with a large amplitude

after its mass becomes much larger than the Hubble param-

eter. Thus, the ALP produced in the FOPT can explain DM

for a broader parameter space which is more accessible to

experiments such as IAXO, ALPS-II, and DM-radio. We also

discuss cosmological aspects of the dark sector that triggers

the FOPT. In particular, we propose a possible solution to

the cooling problem of dark glueballs.

Interestingly, the viable parameter space includes a re-

gion in which the ALP-photon conversion can explain

the observations of very high energy photons from the

extremely bright gamma-ray burst GRB 221009A by the

Large High Altitude Air Shower Observatory (LHAASO)

and Carpet-2. The GRB 221009A was detected by Fermi

GBM and Swift [18,19], and it was accompanied by

O(1000) gamma-rays observed by LHAASO with energy up

arXiv:2210.10022v2 [hep-ph] 31 Oct 2022

to 18TeV [20] and a photon-like airshower of 251 TeV by

Carpet-2 [21]. Such very high energy photons could not

have reached us from the reported redshift z'0.151 [22–

24] because of the efﬁcient electron-positron pair cre-

ation within the extragalactic background radiation. If

the observed very high energy gamma rays are not due

to the Galactic source, this is a hint of the beyond SM

physics. It was discussed that this apparent contradiction

can be resolved if there exists an ALP with mass of or-

der 10−7-8 eV and the ALP-photon-photon coupling gaγγ of

order 10−11 GeV−1[25–27]. In this parameter region, the

abundance of ALP produced by the ordinary misalignment

mechanism is far below the observed DM abundance.1We

will see that, if the ALP is produced from the FOPT, it can

naturally explain DM in this parameter region suggested by

GRB 221009A. See Refs. [37,38] for other phenomenological

implications.

We also discuss the formation of quasi-stable non-

topological solitons called oscillons after the phase transi-

tion. The formation of oscillon requires an O(1) density per-

turbations at the phase transition for the case of the SOPT,

including the case for QCD axion. This is because the am-

plitude of ALP changes adiabatically and its perturbations

cannot grow much during the SOPT. This is in contrast to

the case of FOPT, where the amplitude of ALP oscillation is

as large as its decay constant and the duration of the insta-

bility is long enough for its perturbations to grow exponen-

tially after the phase transition. In other words, the scenario

with the FOPT inevitably results in the formation of oscil-

lons.

ALPs from sudden change of potential. — In this letter,

we denote the ALP mass as ma(T) and its present value as

m0. In the standard scenario, it is usually assumed that the

ALP has a constant mass (ma(T)=m0) from, e.g., a gravita-

tional instanton effect or other explicit symmetry breaking.

The ALP abundance in this case can be calculated as2

ρ(0)

s'45c3/2m1/2

0a2

4π2g(0)

∗sM3/2

Pl Ãπ2g(0)

∗

90 !3/4

, (1)

where aiis the initial ALP ﬁeld value, g∗(g∗s) is the effective

relativistic degrees of freedom for energy (entropy) density,

and MPl '2.4 ×1018 GeV is the reduced Planck mass. The

most natural value of aiis of order the decay constant, fa.

1The ALP abundance is enhanced by the anharmonic effect if the ALP ini-

tially sits near the hilltop by some dynamics [28–32]. However, this di-

rection does not work since the axionic isocurvature perturbation is sig-

niﬁcantly enhanced [33,34]. One exception is when the potential is very

ﬂat near the hilltop. In this case the ALP slow-roll solution is an attractor,

which suppresses the isocurvature perturbations [35]. Another possibil-

ity is to make use of the clockwork mechanism to enhance the ALP cou-

pling to photons [15,36], while keeping its oscillation amplitude large.

2Here and in what follows, the upper indices (0),(1),(2) mean that the vari-

ables are for cases with the standard constant mass, FOPT and SOPT, re-

spectively.

Here g(0)

∗and g(0)

∗sare evaluated at T=T(0)

osc, where the os-

cillation temperature T(0)

osc is determined by cH(T(0)

osc)=m0

with c'3 being a numerical constant. This gives

T(0)

osc 'Ãm0MPl

cs90

π2g(0)

∗!1/2

. (2)

The ALP can have a Chern-Simons coupling to a hidden

SU(N) gauge ﬁeld, in which case it acquires a potential via

the instanton effect. If the conﬁnement phase transition of

SU(N) is of the second order, the ALP has the temperature-

dependent potential such as

ma(T)=m0µT

Tc¶−n/2

, (3)

where Tcis a temperature at which the ALP mass becomes

as large as the present value and nis a critical exponent of

the phase transition. According to the dilute instanton gas

approximation, the exponent is given by n=11N/3+Nf/3−

4 for SU(N) with NFﬂavors, which is consistent with lattice

calculations in several examples. We assume that the ALP

mass becomes constant at T=Tcfor simplicity, though this

does not affect our conclusions. In this case, the ALP starts

to oscillate at cH(T(2)

osc)=m(T(2)

osc). Subsequently, the ALP

mass changes adiabatically, ¯¯˙

m/m2¯¯'(n/4)H(T)/m(T)¿1

for T¿T(2)

osc, and its number density per entropy density

becomes almost constant. Its abundance at present is then

given by

ρ(2)

s'Ãm0

ma(T(2)

osc)!1/2

×ρ(0)

s¯¯¯¯¯g(0)

∗(s)→g(2)

∗(s)

(4)

'45a2

4π2g(2)

∗s

m(n+2)/(n+4)

Tn/(n+4)

c



MPl sπ2g(2)

∗

90 



(n+6)/(n+4)

, (5)

where g(2)

∗(s)=g∗(s)(T(2)

osc). For example, the QCD phase tran-

sition is the second order and its exponent is n'7.84 with

Tc'147MeV for the QCD axion, which is obtained by ﬁtting

the results of lattice simulations [39] up to higher tempera-

ture [40].

Now let us consider the case in which the ALP mass (or its

potential minimum) changes instantaneously via the FOPT.

For deﬁniteness, we consider a simpliﬁed case with the sud-

den change from negligibly small axion mass to a constant

mass m0. In this case, the result is similar to the case with

the constant mass but with independent parameters m0

and T(1)

osc:

ρ(1)

s'45m2

0a2

4π2g(1)

∗s³T(1)

osc´3, (6)

where T(1)

osc is now determined by the temperature at the

FOPT. Here we have assumed cH(T(1)

osc)<m0. If this is not

satisﬁed, the ALP does not start to oscillate at the FOPT. The

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TU-1172Axiondarkmatterfromrst-orderphasetransition,andveryhighenergyphotonsfromGRB221009AShotaNakagawa,1,¤FuminobuTakahashi,1,MasakiYamada,1,2,andWenYin1,§1DepartmentofPhysics,TohokuUniversity,Sendai,Miyagi980-8578,Japan2FRIS,TohokuUniversity,Sendai,Miyagi980-8578,Japan(Dated:November1,2022)Westu...

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TU-1172 Axion dark matter from ﬁrst-order phase transition and very high energy photons from GRB 221009A

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